Beis Shamai and Beis Hillel are arguing here concerning two points: (a) Beis Shamai forbids Olas Re'iyah on Yom Tov while Beis Hillel permits it, and (b) Beis Shamai forbids Semichah on Yom Tov while Beis Hillel permits it.

Their logic is explained in Beitzah (19a-20b):

The first argument revolves around how much Melachah the Torah permitted on Yom Tov. In the second argument, according to one opinion, they are debating whether Shalmei Chovah require Semichah, or according to another opinion, whether Semichah must be immediately before the Shechitah. (See Insights to Chagigah 7b.)

Regarding the philosophical implications of their arguments, in Beitzah (20b) Beis Hillel permits sacrificing Olos on Yom Tov, even though Melachah is normally permitted on Yom Tov only for Ochel Nefesh or for the needs of Yom Tov. Even though we do not need Olos for ourselves (since we do not eat the meat of an Olah), Beis Hillel permits it "in order that our table not be full and Hash-m's table empty." (See Insights to Beitzah 20b about this point.)

If so, perhaps the philosophical point of debate here is that Beis Shamai does not consider offering an Olah as "filling-up" the table of Hash-m, to make it no less than our own table. Beis Shamai maintains that there is no need for us to feel bad when we eat without sacrificing a Korban to Hash-m, since everything that we are doing is based on the command of Hash-m, the Torah. Beis Hillel, in contrast, say that we should look at the matter the way a simple Jew would look at it, who feels something is lacking if he does not bring a Korban to Hash-m on the day that he celebrates with large meals in his own house.

Mordecai Kornfeld